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294                     ANALYTICAL MECHANICS
Since w' is negligible compared with m, and m compared with M, the last equation may be written in the form
m/.P\2 /a_'Y M~-\P'l 'W'
which gives the ratio of the mass of the planet to that of the sun.
225. Kepler's Laws.  In establishing the truth of the law of gravitation Newton showed that the same law which makes the apple fall to the ground keeps the moon in its orbit. Then he extended the application of the law to the other members of the solar system by accounting for the empirical laws which Kepler (1571-1630) had formulated from the observations of Tycho Brahe (1546-1601) . The following are the usual forms in which Kepler's laws are stated,
1.    Each planet describes an ellipse in which th& sun occu-
pies one focus.
2.    The radius vector describes equal areas in equal inter-
vals of time.
3.    The square of the period of any planet is proportional
to the cube of the major axis of its orbit.
The first law is, as we have seen, a direct consequence oi the inverse square law.
The second law follows from equation (III), which holds good for all bodies moving in central fields of force.
The third law amounts to stating that the masses of the planets are negligible compared with the mass of the sun, For if m, a, and P refer to one planet and mf3 a', and Pr tc another planet, then
and p'^
VT (M + m)                V7 (M + m!}
Y = /
'l     \af